A Stiffness-Adjustable Hyperredundant Manipulator Using
a Variable Neutral-Line Mechanism for Minimally Invasive
Surgery
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Citation
Kim, Yong-Jae, Shanbao Cheng, Sangbae Kim, and Karl
Iagnemma. “A Stiffness-Adjustable Hyperredundant Manipulator
Using a Variable Neutral-Line Mechanism for Minimally Invasive
Surgery.” IEEE Transactions on Robotics 30, no. 2 (April 2014):
382–395.
As Published
http://dx.doi.org/10.1109/tro.2013.2287975
Publisher
Institute of Electrical and Electronics Engineers (IEEE)
Version
Author's final manuscript
Accessed
Mon May 23 11:08:33 EDT 2016
Citable Link
http://hdl.handle.net/1721.1/98275
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Creative Commons Attribution-Noncommercial-Share Alike
Detailed Terms
http://creativecommons.org/licenses/by-nc-sa/4.0/
A Stiffness-Adjustable Hyper-Redundant Manipulator using a
Variable Neutral-line Mechanism for Minimally Invasive Surgery
Yong-Jae Kim, Shanbao Cheng, Sangbae Kim, and Karl Iagnemma
Abstract— In robotic single port surgery, it is desirable for a
manipulator to exhibit the property of variable stiffness. Small
port incisions may require both high flexibility of the
manipulator for safety purposes, and high structural stiffness
for operational precision and high payload capability. This
paper presents a new hyper-redundant tubular manipulator
with a variable neutral-line mechanisms and adjustable
stiffness.
A unique asymmetric arrangement of the tendons and the
links realizes both articulation of the manipulator and
continuous stiffness modulation. This asymmetric motion of
the manipulator is compensated by a novel actuation
mechanism without affecting its structural stiffness.
The paper describes the basic mechanics of the variable
neutral-line manipulator, and its stiffness characteristics.
Simulation and experimental results verify the performance of
the proposed mechanism.
Index Terms - Variable neutral-line mechanism, snake–like
manipulator, adjustable stiffness, medical robot.
(a)
I. INTRODUCTION
S
NAKE-like manipulators have unique characteristics
and advantages such as flexibility, safety, dexterity, and
potential for minimization. Most snake-like manipulators
can be roughly categorized into flexible manipulators and
(b)
hyper-redundant manipulators. Trunk and tentacle-like
Fig.1.
(a)
4-DOF
variable
neutral-line
manipulator and actuation system,
devices made from soft materials belong to the flexible
(b) Single-port surgical system using the variable neutral-line
manipulator category [1, 2, 3, 4] and they have inherent
manipulator
passive compliance, which is one of the great advantages of
schemes. This compliance is caused by flexibility of the
these manipulators. On the other hand, hyper-redundant
structural material, or deformable components in the
manipulators [5, 6, 7] are composed of many rigid links and
mechanism. Because the compliance is determined by the
joints, which can be actuated by embedded motors as in [7],
material stiffness and the pose of the manipulators [2, 8], the
or, by external actuators and transmission components such stiffness remains fixed if the pose is determined.
as tendons or flexible shafts [5, 6]. Most of the externally
Recently, snake-like manipulators are receiving high
actuated hyper-redundant manipulators have an under
actuated property, and thus they also exhibit passive attention due to rising research interest in minimally
invasive surgery (MIS) and natural orifice translumenal
compliance like flexible manipulators.
Passive compliance is useful for safe manipulation of endoscopic surgery (NOTES) [9, 10, 11]. MIS and NOTES
unknown or delicate objects, human-robot interaction, and have huge advantages including low trauma, fast healing and
multi-arm cooperation without complex force-feedback minimal or no scarring. Snake-like devices are well suited
for accessing deep inside of the patient’s abdominal cavity
Manuscript received March 10, 2012. This work was supported in part
through a small entry point, while today’s rigid (and
by SAIT of Samsung Electronics.
typically straight) tools have difficultly achieving such
Y.J. Kim is with Samsung Advanced Institute of Technology, Ki-hung
access. However, the fixed stiffness of snake-like devices
gu, Yong-in si., Kyunggi do, Rep. of Korea. (corresponding author to
provide phone: +82-31-280-1528, e-mail: yj424.kim@samsung.com)
hinder their ability to achieve high stiffness for high payload
S. Cheng is with the Ellipse Technologies, Irvine, CA, 92618, USA
operation and exact positioning, and low stiffness for safe
(phone:+1-585-643-4095; e-mail: chengshanbao@gmail.com).
movement without harming internal organs [12, 13]. In order
S. Kim and K. Iagnemma are with Massachusetts Institute of
Technology (MIT), Cambridge, MA, 02139, USA (phone:
to overcome this drawback, various stiffening mechanisms
+1-617-452-2711, e-mail: sangbae@mit.edu and phone: +1-617-452-3262,
have been developed. One popular approach relies on the
e-mail: kdi@mit.edu)
use of wire tension and friction between rigid links [5, 14].
1
Length invariant
neutral-line
Flexible
backbone
or
pivots and
springs
θ
F
− Δl
+ Δl
+ Δl
− Δl
w
+ Δl
− Δl
(a)
(b)
(c)
Fig.2. One dimensional illustration of traditional snake-like
manipulators
However, the wire tension must be very high, and thus the
links must be strong enough to endure high tension, since the
stiffening force arises solely from the friction between the
links. Moreover, the links can occupy substantial space, and
thus it is difficult to create a compact manipulator that also
allows passage of miniaturized surgical tools (e.g. one with a
large hollow space at the center).
Recent research has focused on the use of
tunable-stiffness materials, including field-activated
materials
like
magnetorheological
(MR)
or
electrorheological (ER) fluids [15, 16]. These technologies
are promising for precise control of damping, and are mostly
used in active damping mechanisms such as tunable
automotive suspensions. However, there are limitations in
their achievable range of the elastic modulus, or yield
strength, when they are activated. Thermally activated
materials such as wax or solder can also be used as
tunable-stiffness elements to create locking mechanisms in
soft robotic applications [17, 18]. However, achieving
tunable stiffness with these materials requires long activation
times (typically on the order of seconds).
Particle jamming technology using granular media has
also recently been researched as another way to achieve
tunable stiffness [19, 20]. Particle jamming has interesting
features including high deformability in its fluid-like state,
and drastic stiffness increase in its solid-like state, without
significant change in volume, but it requires substantial
volume to achieve sufficient stiffness. In order to overcome
this problem, reduced dimensional jamming approaches,
such as layer jamming of thin frictional flaps of material,
have been investigated and exhibited drastic stiffness
changes within a small volume [21,22].
To overcome the drawbacks of previous methods, we
propose a new snake-like mechanism, named a variable
neutral-line mechanism as shown in Fig.1, whose stiffness
can be changed continuously, even while in motion.
Although it is also composed of rigid links and actuated by
tendons, its unique joint mechanism and actuation system
make stiffness control possible by varying the tension of the
tendons. Its simple, thin, and hollow structure (top left of
Fig.1(a)) is suitable for surgical application such as MIS or
NOTES, because it can be used as a flexible guide tube
through which multiple miniaturized surgical tools such as
endoscopes and surgical instruments can be inserted. Fig.1(b)
shows a single port surgical system which is composed of
dual 7-DOF instruments and one 3-DOF endoscopic device
mounted at the end of a guide tube made by the two identical
2-DOF variable neutral-line manipulators. In aid of the
rotation and insertion motion of the mounting system, the
guide tube has in total 6-DOF, which means that the guide
tube can place the dual instruments and the endoscope at
arbitrary position in the abdominal cavity with an arbitrary
approach direction. Then, the human-like configuration of
the instruments and the endoscope can provide an intuitive
teleoperational environment for a surgeon. Due to dexterity
and adjustable stiffness of the variable neutral-line
manipulator, extremely difficult surgeries using a single port
(such as lower anterior resection or partial nephrectomy) can
be performed significantly more easily and safely.
The paper is structured as follows. In Section II, we
introduce the basic mechanics of the proposed manipulator
without external forces, and then, in Section III, we
exhaustively investigate the deflection shape under an
external force. Based on this, Section IV derives important
stiffness properties of the variable neutral-line manipulator.
Section V presents an actuation system that can control both
of the bending angle and the stiffness of the manipulator
independently. Section VI validates the stiffness model
through experiments. Section VII discusses key design
parameters related to the stiffness, and suggests several
useful design variations without losing the advantage of the
proposed mechanism, and this is followed by the paper's
conclusion.
II. BASIC MECHANICS OF VARIABLE NEUTRAL-LINE
MANIPULATOR
A one-dimensional concept of a traditional snake-like
manipulator is shown in Fig.2. This structure is composed of
a flexible backbone at the center with evenly placed disks
(the proximal part of Fig.2(a)) [1], or multiple links with
pivots at their centers, with springs fixed around them (the
distal part of Fig.2(a)) [6]. They are actuated by a pair of
wires placed at an equal distance from the center. Fig.2(a)
illustrates an example of the bent pose when the left-side
wire is pulled and the other is loosened. If we assume that
the compression of the backbone (or pivot mechanism) is
negligible compared to bending, the length of the centerline
is invariant. The relationship between amount of the
movement of the right and left wires,
bending angle of the manipulator,
as follows:
Δl r = − Δll =
w
θ.
2
θ
Δl r
and
Δl l , and the
, can be easily obtained
(1)
2
where w denotes the width between the left and right wires.
This equation shows that the left and right wires have the
same magnitude and opposite direction. This symmetric
movement of wires provides an under actuated mode that
introduces passive compliance, as mentioned in the previous
section. When we consider a situation where the wires are
fixed at a straight pose, as in Fig.2(b), and an external force
is applied to the lateral direction at the end of the
manipulator, it tends to bend easily into to an s-shape
regardless of wire tension, as shown in Fig.2(c). This is
because the manipulator can be moved to various
configurations without changing the wire length. Therefore,
the manipulator stiffness to resist bending comes solely from
the fixed stiffness of the flexible backbone, or the springs,
and not from wire tension.
Length invariant
neutral-line
θ
F
Δl −
A. 1 DOF Variable Neutral-line Manipulator
Δl +
Δl +
Δl
−
w
Δlr
Tension
Δl − + Δl + > 0
− Δll
Tension
(a)
(b)
(c)
Fig.3 1-dimensional concept of the variable neutral-line
manipulator
Δφ
d r ( Δφ )
Δφ + α
d l ( Δφ )
w
r − r cos(α + Δφ )
(a)
From the geometry of each joint, the relationship between
the bending angle of the manipulator and the movement of
the wires can be derived. If we assume that there is no
external force, the bending shape will be arc shapes, as in
Fig.3(a). Let us define n and Δφ as the number of joints
(not links), and the bending angle of one joint with respect to
the proximal link. As already defined, θ is the bending angle
of the manipulator tip, and as the manipulator has an arc
shape, we know that θ = nΔφ . Fig.4 illustrates a close-up
view of the rolling joint, where
r
α
implies that proper tension of the wires dictates the force
maintaining the shape of the manipulator. (A detailed
analysis of the relationship between wire tension and
stiffness of the manipulator will be investigated in Section
III and IV.)
In contrast to traditional snake-like
manipulators, the location of the length invariant neutral-line
shown in Fig.3 is not fixed at the center. In mechanics,
neutral-line (or neutral surface) denotes a conceptual line
within a beam or cantilever, where the there is no
compression or tension and thus the length does not change.
As the position of the neutral-line varies according to the
pose of the proposed mechanism, this device is termed a
variable neutral-line manipulator.
The contact surface of the joint does not need to have a
circular shape. Many other asymmetric joints can produce
the variable stiffness property, but we will here focus on
cylindrical rolling joint for simplicity, and discuss other
possibilities briefly in Section VII. In the remaining part of
this section, the basic mechanics and the asymmetric
property in the situation of no external forces are described.
(b)
Fig.4 Close-up view of the 1-dimensional rolling joint
Moreover, in order to obtain higher stiffness, stiffer
materials must be used and, accordingly, a higher wire
tension is needed to overcome the stiffness of the backbone
or springs, which can lead to a reduction of available drive
actuator force and a larger required wire diameter .
On the other hand, if asymmetric movement of the wire
pair can be achieved, the stiffness of the manipulator can be
a function of the wire tension. Fig.3 shows an example of an
asymmetric mechanism. This mechanism employs
self-similar links with rolling joints, instead of pivots or a
backbone, where the joint has arc-shape contact surfaces.
When it bends similar to Fig.3(a), movement of the loosened
wire Δlr is longer than that of pulled wire Δl l , which leads
to asymmetric behavior. When a lateral force is applied, as
shown in Fig.3(c), the total movement of the wires
Δl − + Δl + has a positive value. This is important, since it
r , w , α , d l and d r
represent the radius of contact surface, the width between
the left and right wires, half of the contact angle, and local
length of the left and right wires, respectively. From Fig.4(b),
d l and d r can be calculated as follows:
Δφ ⎞
⎛
d l ( Δφ ) = 2r ⎜1 − cos(α −
) ⎟
2 ⎠
⎝
Δφ ⎞
⎛
d r ( Δφ ) = 2r ⎜1 − cos(α +
) ⎟
2 ⎠
⎝
(2)
Therefore, the movements of the left and right wires, Δd l
and Δd r , are
Δφ ⎞
⎛
Δd l ( Δφ ) ≡ d l ( Δφ ) − d l (0) = 2r ⎜ cos α − cos(α −
) ⎟
2 ⎠
⎝
Δφ ⎞
⎛
Δd r ( Δφ ) ≡ d r ( Δφ ) − d r (0) = 2r ⎜ cosα − cos(α +
) ⎟
2 ⎠
⎝
(3)
Consequently, the relationship between θ and the overall
length change of the left and right wire,
Δl l
and
Δlr ,
without any external force, can be obtained as (4). Here we
can notice that the sum of the length change is nonzero.
3
θ ⎞
⎛
Δll (θ ) = nΔd l = 2nr ⎜ cosα − cos(α − ) ⎟
2n ⎠
⎝
θ ⎞
⎛
Δlr (θ ) = nΔd r = 2nr ⎜ cosα − cos(α + ) ⎟
2
n ⎠
⎝
(4)
B. 2 DOF Variable Neutral-line Manipulator
Fig.5 illustrates a diagram of a 2-degree of freedom
manipulator. Each link shown in Fig.5(a) has two cylindrical
contact surfaces, oriented orthogonally to each other. At the
center of the link there is a large hole, which distinguishes
the device structurally from many other snake-like
manipulators. The contact surfaces between adjacent links
form rolling joints, and they are alternately placed at
orthogonal directions to each other as shown in Fig.5(b) and
(c). In order to prevent slip between the rolling surfaces,
flexures can be used which will be described in Sect V and
Fig.14(b).
In the case of the 2-DOF manipulator, as shown in
Fig.6(b), there are two wire pairs, and the motion of the two
pairs affects each other. In other words, when the
manipulator is deflected by one wire pair’s motion, the other
wires placed at the center of links (the red line in Fig.6(b))
are also pulled. Let us call the horizontal motion and vertical
motion in Fig.6(a) pan motion and tilt motion, respectively,
and let the corresponding wire pairs be called the pan wire
pair and tilt wire pair. The manipulator has the same number
of links n for each pan motion and tilt motion. Also, Δ φ p
and Δφt denote bending angles for pan motion and tilt
motion of a joint, and Δd pl , Δd pr , Δd tl and Δd tr denote
displacements of the lengths of the pan wire pair and tilt
wire pair of a joint, respectively. In a similar way as (3), the
displacement of the pan wires can be calculated as follows:
contributes to the movement of the pan wires. It can be
derived from the diagram of the rolling joint in Fig.6(b). In a
similar way as (4), the relationship between the angle of the
end effector and the length change of the pan wires can be
obtained as:
θ
⎛
θ ⎞
Δl pl (θ p ,θt ) = 2nr ⎜⎜ cosα − cos(α − p ) + 1 − cos t ⎟⎟
2
n
2
n ⎠
⎝
θ
⎛
θ ⎞
(6)
Δl pr (θ p ,θ t ) = 2nr ⎜⎜ cosα − cos(α + p ) + 1 − cos t ⎟⎟
2
n
2
n ⎠
⎝
where Δl pl , Δ l pr , θ p and θ t denote
the
overall
displacement of the left pan wire and right pan wire, and the
components of the end effector tip angle in the directions of
pan motion and tilt motion, respectively. These equations are
valid when an external force does not exist, and the bending
angle of each joint has the same angle. The relationship
between the angle of the end effector and the length change
of the tilt wires also can be calculated in the same way.
III. DEFLECTION CHARACTERISTICS UNDER EXTERNAL
FORCE
In order to show that the stiffness of the manipulator is
controllable by changing the wire tension, the bending shape
of the manipulator under an external force will be calculated.
In this paper, small deflections about the straight pose will
be considered. The deflection diagrams (such as Fig.3(c) and
Fig.7(a)) show an exaggerated shape for ease in explanation.
Since we assume that the manipulator links are rigid, an
axial force applied to the tip while at a straight pose causes
no effect, and thus only lateral forces cause deflection.
Analysis of the manipulator stiffness under an arbitrary
bending angle, and the effect of moment loading at the end
effector, are both beyond the scope of this paper. Therefore,
we will focus on analysis of manipulator deflection when
subject to a lateral force.
Let us start with analysis of a 1-DOF manipulator. As
shown in Fig.7(a), the amount of lateral motion d generated
by a lateral force in the y direction is:
n
d = ∑ l sin φi
(7)
i =1
Fig.5. 2-dimensional concept of the variable neutral-line
manipulator
Δφ p
⎛
Δφ ⎞
Δd pl ( Δφ p , Δφt ) = 2r⎜⎜ cosα − cos(α −
) + 1 − cos t ⎟⎟
2
2 ⎠
⎝
Δ
φ
⎛
Δφ ⎞
Δd pr ( Δφ p , Δφt ) = 2r ⎜⎜ cosα − cos(α + p ) + 1 − cos t ⎟⎟
2
2 ⎠
⎝
(5)
The term 1 − cos(Δφt / 2) in (5) implies that the tilt motion
4
subject to an external force, and thus angle φi in each link is
different from each other. Thus, Δll and Δlr should be
calculated by summing the displacement of the wires of each
link using (3).
n
n
Δφ ⎞
⎛
Δll = ∑ Δd l ( Δφi ) = ∑ 2r ⎜ cos α − cos(α − i ) ⎟
2 ⎠
⎝
i =1
i =1
n
n
Δ
φ ⎞
⎛
(10)
Δl r = ∑ Δd r ( Δφi ) = ∑ 2r ⎜ cos α − cos(α + i ) ⎟
2 ⎠
⎝
i =1
i =1
where Δφi = φi − φi −1 , φ0 = 0 and φn = 0 .
If we substitute (10) into (8) we find:
n
Δφ ⎞ n ⎛
Δφ ⎞
⎛
∑ 2r⎜ cosα − cos(α − i ) ⎟ = ∑ 2r⎜ cosα − cos(α + i ) ⎟
2 ⎠ i =1 ⎝
2 ⎠
⎝
Δφi
Δφi ⎞ n ⎛
Δφ ⎞
⎛
) − cos(α −
) ⎟ = ∑ ⎜ − 2 sin(α ) sin( i ) ⎟ = 0
⎜ cos(α +
∑
2
2
2 ⎠
⎠ i =1 ⎝
i =1 ⎝
Because sin α is nonzero, (8) is reduced to
i =1
n
n
∑ sin(
i =1
Fig.6. (a) Pan motion and Tilt motion (b) Close-up view of tilting
surface of 2-dimensional variable neutral-line manipulator
y
y
Lateral force
l
φn −1 φn = 0
d
φ3
φ3
φ2
φ1
φ1
(a)
x
φ2
d
2
(b)
x
Fig.7. Deflection model of the straight pose 1-DOF manipulator
under a lateral force
where l , n and φi denote the length of the links, number of
th
joints and the bending angle of the i joint with respect to
the x axis. Note that φn is always zero. In the case of large
deflections, the joints move differently from ordinary
revolute joints, however since we only consider small
motion, we approximate them as revolute joints. As the two
wires are pulled to maintain a straight pose (refer to Fig.3(c)),
the movements of the left and right wires are the same, i.e.
Δll = Δlr .
(8)
The objective is to find the pose of the manipulator under
lateral motion that minimizes the change of length of the
wires. Therefore the problem is to find the set of φi as
follows:
arg min(Δll + Δlr )
(9)
φ
where φ denotes the set of φi , and thus it represents the
deflected pose of the manipulator. Δll and
Δlr in (8) and (9)
have different values than in (4) because the manipulator is
Δφi
) = 0.
2
(11)
Also, the cost function Δll + Δlr in (9) can be simplified
(using (10)) as
n
Δφ
Δφ ⎞
⎛
Δll + Δlr = 4nr cos α − ∑ 2r ⎜ cos(α + i ) + cos(α − i ) ⎟
2
2 ⎠ (12)
⎝
i =1
n
Δφ
= 4nr cos α − 4r cos α ∑ cos( i )
2
i =1
Thus the minimization problem (9) can be rewritten by a
maximization problem. Consequently, we can obtain the
optimization problem of (13) with the two constraints (7)
and (11) as follows:
Δφ ⎞
⎛ n
arg max ⎜ ∑ cos( i ) ⎟
2 ⎠
φ
⎝ i =1
(13)
such that
n
∑ sin φ
i
i =1
=
d
and
l
n
∑ sin(
i =1
Δφi
) = 0.
2
For a more simplified formulation, it is worthwhile to
consider the geometric property of a tendon-supported
structure under lateral forces. The lateral force as shown in
Fig.7(a) generates a torque, or moment, at the proximal end
of the manipulator, and the wire pair delivers the same
amount of torque to the distal end [2]. As a result, the
manipulator deflects into a symmetric s-shape, because the
same amount of torque is applied at each end. Exploiting this
property, the problem can be represented by half of the joint
configuration of the manipulator, as shown in Fig.7(b). This
symmetric property can eliminate the constraint of (11),
because the equality of the displacement of both wires (8) is
Δφ
φ
i is the change of i as shown in
automatically satisfied.
Equation 10, and its sign could be either positive or negative
depending on its location. For example, in Fig.7(a), the
lower half of the manipulator will always have positive
Δφi , while the upper half will have negative Δφi .
5
A. Analysis of Manipulators with Odd Number of Joints
Therefore, in the case of an odd number of joints, the
optimization function (13) and constraint (7) are rewritten
as:
where we call the m + 1 by m + 1 matrix on the left side A ,
and the m + 1 vector on the right side c . In the case of 3-joint
manipulators (i.e. n = 3 ), A and c are:
m −1
Δφ
Δφ ⎞
⎛ 1
arg max ⎜ cos( m ) + ∑ cos( i ) ⎟
2
2 ⎠
φ
i =1
⎝ 2
A n =3
(14)
such that
m −1
∑ sin φ
i
i =1
(15)
where Δφi = φi − φi −1 ,
⎡φ ⎤
−1
⎢λ ⎥ = A c
⎣ ⎦
φ0 = 0 and m = (n + 1) / 2 .
Due to the assumption of small deflections, the
trigonometric function in (14) and (15) can be approximated
using a Taylor series expansion, as
m −1
m −1
1
Δφ
Δφ
1 ⎛ 1
⎛ 1 2 ⎞
2 ⎞
cos( m ) + ∑ cos( i ) = ⎜1 − Δφm ⎟ + ∑ ⎜1 − Δφi ⎟
2
2
i =1
2
2 ⎝
8
⎠
i =1
⎝
8
⎠
m −1
1 ⎞ 1 ⎛ 1
⎛
2
2 ⎞
= ⎜ m − ⎟ − ⎜ Δφm + ∑ Δφi ⎟
2
8
2
⎝
⎠
i =1
⎝
⎠
∑ sin φi = ∑
i =1
4
In order to solve this approximate optimization problem,
the method of Lagrange multipliers is applied. The Lagrange
function is defined as:
(16)
Λ(φ ) = f (φ ) + λg (φ )
where
f (φ ) =
m −1
1
⎛ m ⎞ d
2
2
Δφm + ∑ Δφi , g (φ ) = ⎜ ∑φi ⎟ −
2
i =1
⎝ i =1 ⎠ 2l
λ is
a Lagrange multiplier. f (φ ) again represents the
minimization function, because the sign is changed during
approximation. Differentiating (16), the following necessary
conditions for the solution are obtained.
4φi − 2φi +1 + λ = 0,
i =1
⎧
⎪− 2φ + 4φ − 2φ + λ = 0, 1 < i < m − 1 (17)
∂Λ (φ ) ⎪
i −1
i
i +1
= ⎨
−
2
φ
+
3
φ
−
φ
i = m −1
∂φi
i
−
1
i
i
+
1 + λ = 0,
⎪
⎪⎩
− φi −1 + φi = 0,
i=m
∂Λ (φ ) m−1
d
= ∑ φi − = 0
∂λ
2
l
i =1
These can be expressed in matrix form as
1⎤
⎡ 4 − 2
⎢ − 2 4 − 2
1⎥
⎢
⎥
−2 4 −2
1⎥
⎢
⎡ 0m ×1 ⎤
⎢
⎥
" " "
! ⎥ ⎡φm ×1 ⎤ ⎢
⎥
⎢
=
⎢
⎥ ⎣d / 2l ⎦
⎢
− 2 3 − 1 1⎥ ⎣ λ ⎦
≡c
⎢
⎥
− 1 1 0⎥
⎢
⎢⎣ 1
1
1
...
1
0 0⎥⎦
≡A
(19)
n=6
4
5
4.5
3
3
2
2
4
1
1
3.5
0
0
y5
20
40
60
80
0
x
0
3
2
2
60
80
x
n = 30
4
3
40
2
20
40
60
80
x
0
n=3
n=6
n = 10
n = 30
1.5
1
0.5
1
0
3
2.5
y5
n = 10
20
0
20
40
60
80
x
0
0
10
20
30
40
(a)
i =1
and
5
n=3
0
φi .
y 5.5
y
5
1
m
.
(21)
y
4
and
m
⎡ 0 ⎤
c = ⎢ 0 ⎥
⎢
⎥
⎢⎣d / 2l ⎥⎦
and
(20)
As a result, the deflected pose (i.e. the set of φi ) can be
obtained from the following equation.
d
2l
=
⎡ 3 − 1 1⎤
= ⎢− 1 1 0⎥
⎢
⎥
0 0⎥⎦
⎢⎣ 1
(18)
50
60
70
80
90
x
(b)
Fig.8. Deflected shapes of simulated manipulators
B. Analysis of Manipulators with Even Number of Joints
Until now, we have considered the case of manipulators
with an odd number of joints. In the case of an even number
of joints, the symmetric property can be used similarly, if we
take into account that the distal link of the simplified model
has half the size of the other links.
In a similar derivation as (14) and (15), the optimization
problem for a manipulator with an even number of joints is:
Δφ ⎞
⎛ m
arg max ⎜ ∑ cos( i ) ⎟
2 ⎠
φ
⎝ i =1
(22)
such that
m −1
1
d
sin φm + ∑ sin φi =
2
2l
i =1
(23)
where Δφi = φi − φi −1 ,
φ0 = 0 and m = n / 2 .
The equation (21) can be used to find a solution for an
even number of joints, where the matrix A takes on unique
values, as:
1 ⎤
⎡ 4 − 2
⎢ − 2 4 − 2
1 ⎥
⎢
⎥
# # #
" ⎥
⎢
A even links = ⎢
⎥
− 2 4 − 2 1 ⎥
⎢
⎢
− 2 − 2 1 / 2⎥
⎢
⎥
1 ! 1 1 / 2 0 ⎦
⎣ 1
(24)
In the case of 4-joint manipulators (i.e. n = 4 ), A is:
6
A n =4
Table I
⎡ 4 − 2 1 ⎤
= ⎢− 2 − 2 1 / 2⎥ .
⎢
⎥
⎢⎣ 1 1 / 2 0 ⎥⎦
DESIGN-INDEPENDENT STIFFNESS COEFFICIENT
n
In the case of 2-joint manipulators (i.e. n = 2 ), the set of
φi
can be directly calculated from (7) as φ1 ≅ d / l and φ2 = 0 .
Fig.8 shows the deflected shapes of various simulated
manipulators according to their number of links. The length
of the manipulator nl , radius of contact surface r , half of
the contact surface angle α , and amount of the lateral
displacement d are 87mm, 14.8mm, 33.18deg and 5mm,
respectively.
C. 2-DOF Analysis
considered. Let us consider a situation of a 2-DOF
manipulator where the external force in the pan direction
results in a deflecting motion in the same direction. Then,
the displacements of pan wires are the same as (10) because
the tilt angle of each joint is zero. However, the
displacements of the tilt wires are nonzero due to the
coupling term, as follows
n
Δφ ⎞ n ⎛
Δφ ⎞
⎛
Δφ
Δltl = ∑ 2r⎜⎜ cosα − cos(α − ti ) + 1 − cos pi ⎟⎟ =∑ 2r⎜⎜1 − cos pi ⎟⎟
2
2
2 ⎠
i =1
⎝
⎠ i =1 ⎝
n
n
Δφ ⎞
Δφ ⎞
⎛
⎛
Δφ
Δltr = ∑ 2r ⎜⎜ cosα − cos(α + ti ) + 1 − cos pi ⎟⎟ =∑ 2r ⎜⎜1 − cos pi ⎟⎟
2
2 ⎠ i =1 ⎝
2 ⎠
i =1
⎝
(25)
denote displacements of the
left and right tilt wires and the pan and tilt angle of i th link,
respectively, and Δφti is zero because there is no tilt motion.
Here, the objective is to find the pose of the manipulator (i.e.
the set of φ pi ) when subjected to lateral loading that
minimizes the change of lengths of the tilt wires as well as
that of the pan wires. If we consider the minimization
problem for only the tilt wires:
arg min(Δltl + Δltr )
4
5
6
10
30
∞
16.00 13.50 12.80 12.50 12.34 12.12 12.01 12.00
Tension to
tilt wires
Tension to
pan wires
T
F
T
External force
2-DOF manipulator
IV. STIFFNESS OF THE VARIABLE NEUTRAL-LINE
MANIPULATOR
In this section, the relationship between the stiffness of the
manipulator and the wire tension is calculated by using the
deflected shape obtained in the previous section. Fig.9
shows a schematic of the wire tensions and force exerted on
the manipulator, where the same tension T is applied to the
pan wires and tilt wires individually, and F is the external
lateral force applied to the end effector. The tensioning
mechanism is highly simplified here, and the detailed
structure will be described in Section V.
For clarity of the stiffness calculations, let us first define
several terms. In (21), only c contains design parameters l
and d . Thus, we can cancel out these design parameters by
defining the following variable:
si =
nl
h
φi = φi
d
d
(28)
where h = nl and can be considered as the total length of
the manipulator. The set of si can be calculated by
substituting φi in (21) with (28), i.e.
⎡ si m×1 ⎤
−1 ⎡ 0m×1 ⎤
⎢
⎥ = A ⎢n / 2⎥
⎣
⎦
⎣ λ ⎦
φp
(26)
From (25) and (26), the optimization problem is rewritten as
Δφ pi ⎞
⎛ n
arg max ⎜⎜ ∑ cos(
) ⎟
2 ⎟⎠
φp
⎝ i =1
3
Fig.9. A schematic of tensions and force exerted in a
In order to extend these analyses to a 2-DOF manipulator,
the coupling term 1 − cos(Δφt / 2) in (5) should be
where Δltl , Δl tr , Δφ pi and Δφti
Sn
2
.
(27)
Note that (27) for the tilt wires has exactly the same cost
function as that of the pan wires (13). This is important
because it implies that displacing the tilt wires under lateral
loading produces the same deflected pose as displacing the
pan wires. It also means that (13) represents the optimization
problem for the 2-DOF manipulator. Therefore, the deflected
shape obtained from (21) can be applied directly to the
2-DOF manipulator.
(29)
which shows that si is a design-independent constant related
only to the number of the joints n . Using this relationship,
we can define another constant:
n
Sn ≡ n ∑ Δsi
2
i =1
(30)
where Δsi = si − si −1 , s0 = 0 . The values of S n shown in
Table I were calculated by using (29) and (30) for each case
of n . They are also design-independent parameters only
related to n , regardless of the link length l , radius of
contact surface r , or half of the contact surface angle α .
7
Now it is possible to derive the stiffness of the variable
neutral-line manipulator. From (12), the displacement of the
pan wires Δl p can be expressed as follows:
Δl p ≡ Δl pl = Δl pr = ( Δl pl + Δl pr ) / 2
n
Δφi
)
2
i =1
n
⎛ 1 2 ⎞ r cosα n
2
≅ 2nr cosα − 2r cosα ∑ ⎜1 − Δφi ⎟ =
Δφi . (31)
∑
8
4
⎠
i =1 ⎝
i =1
By substituting (28)-(30) into (31), the relationship
between d and Δl p can be calculated as
= 2nr cos α − 2r cos α ∑ cos(
Δl p =
2
⎞
r cos α n ⎛⎜ ⎛ d ⎞
r cos α 2 .
2
Δsi ⎟ = Sn
d
⎜
⎟
∑
⎟
4 i =1 ⎜⎝ ⎝ h ⎠
4nh 2
⎠
)
N
( 25
e
c
r
o
F
(32)
)
N
( 2.5
e
c
r
o
F
y = 0.481x
2
20
y = 0.374x
15
1.5
10
1
5
Tension 20N
Tension 33.1N
Tension 46.5N 0.5
Tension 59.9N
y = 0.266x
0
0
10
20
30
40
(a)
Displacement
(mm)
50
60
0
y = 0.1611x
0
1
2
3
(b)
4
5
6
Displacement
(mm)
Fig.10. Stress-strain graph as determined by simulation. (a)
Overall shape of graph. (b) Close-up shape near zero
TABLE II
SPECIFICATIONS AND OF THE 2-DOF MANIPULATOR
Specification
Value
Number of Joints n
3
Half of Contact angle α
0.579 rad
Radius of Contact surface r
14.8 mm
29 mm
Link Length l
Pretension (N)
20.0, 33.1, 46.5, 59.9 N
TABLE III
STIFFNESS COMPARISON
Tension (N)
20.0
33.1
Simulated Stiffness (N/mm)
0.161
0.266
Calculated Stiffness (N/mm)
0.162
0.268
46.5
0.374
0.376
59.9
0.481
0.484
From (25), the displacement of the tilt wires Δlt also can
be expressed as follows:
n
Δφ ⎞
⎛
Δlt ≡ Δltl = Δltr = 2r ∑ ⎜1 − cos( i ) ⎟
2 ⎠
i =1 ⎝
2
n
⎞
r n ⎛ ⎛ d ⎞
r
⎛ 1
2 ⎞
2
≅ 2r ∑ ⎜ Δφi ⎟ = ∑ ⎜ ⎜ ⎟ Δsi ⎟ = Sn
d2
2
⎜
⎟
8
4
h
4
nh
⎝
⎠
⎝
⎠
i =1
i =1 ⎝
⎠
(33)
By using the virtual work concept, an energy conservation
equation can be obtained as follows:
By putting (32) and (33) into (34) and differentiating both
sides of (33) by d ,
r cos α
r
.
F = T Sn
2d + T S n
2d
2
4nh
4nh 2
(36)
As a result, the stiffness K is obtained as
F
r(1 + cos α )
K = = Sn
T
d
2nh 2
(37)
This verifies that the stiffness of the manipulator can be
adjusted via the tension of the wires, and the relationship
between the stiffness and the tension is approximately linear.
This is an important advantage of the variable neutral-line
design compared to other comparable manipulators. Note
that increasing the number of joints reduces the manipulator
stiffness because n is in the denominator of (37). However,
if the maximum bending angle is considered, this leads to a
different conclusion, which will be discussed in Subsection
VII.B.
Fig.10 illustrates the simulation results of the manipulator
with the specification of Table II, where frictional effects are
ignored. For the simulation, the multi-body dynamics
simulation software MSC ADAMS 2012 was used. When
the maximum lateral motion in Fig.10(a) is extremely high
(more than 60mm in minimum tension), the overall graph
does not follow a linear relationship because the manipulator
behavior lies outside the valid range of the Taylor series
approximations applied to (14), (15), (22) and (23). However,
in a reasonably small region about zero displacement, it
shows an approximately linear relationship.. Table III shows
the simulated stiffness and calculated stiffness using (37).
The error between them is negligible, which means that the
stiffness calculation and approximation is valid.
V. ACTUATION SYSTEM DESIGN
Most current tendon-driven mechanisms employ pulley
mechanisms or linear actuators, due to the symmetric
property of the wire pair. Fig.11(a) shows a concept of an
actuator design using a pulley mechanism. This simple
approach cannot be easily applied to the variable neural-line
manipulator because it has an asymmetric wire movement. If
the pulley mechanism is used for the variable neutral-line
manipulator, the pulley is pulled as it deflects, which means
that the pivot of the pulley moves in a wide range, and the
wire tension affects the torque of the actuator when it bends.
Alternatively, two linear actuators can be used to actuate two
wires individually, but these actuators must have a complex
force control function for tension control.
d
∫ Fdx = TΔl
p
+ TΔlt
0
(34)
8
Δγ l and Δγ r
Manipulator side
Actuator side
β ψ
2
−
2
γ r (ψ )
γ l (ψ )
Manipulator side
Actuator side
Rotational
Actuator
β
β
Pulley
Rotational
Actuator
Pan shape
Lever
ra
ψ
Pretension
(a)
(b)
Fig.11. Conceptual Design of Actuating Mechanisms
(a) Conventional concept (b) Proposed concept
Tilt motion
Pan motion
Manipulator side
are
β
β ψ ⎞
⎛
Δγ l (ψ ) = γ l (ψ ) − γ l (0) = −2ra ⎜ cos( ) − cos( − ) ⎟
2
2 2 ⎠ (39)
⎝
β
β ψ ⎞
⎛
Δγ r (ψ ) = γ r (ψ ) − γ r (0) = −2ra ⎜ cos( ) − cos( + ) ⎟
2
2 2 ⎠
⎝
Similarity between (4) and (39) shows that the proposed
actuation mechanism can compensate the asymmetric
motion of wire pair. If we set the length of lever ra as nr
and half of the pan shape angle β as 2α , the rotation of the
lever and the bending angle of the manipulator has a direct
relation ψ = θ / n . Thus, the rotational actuator can solely
control the bending motion of the manipulator, and the
pretension shown in Fig.11(b) can adjust the stiffness
independently.
This actuation mechanism is for a 1-DOF manipulator. For
complete compensation of 2-DOF motion as (6), another
mechanism is required. Fig.12 shows the actuation
mechanism for a 2-DOF manipulator. The length of the
brown colored wire changes as the lever rotates, as follows:
Δν (ψ p ) = 2ra cos
ψp
2
(40)
Actuator side
Similar to (39), it can compensate the last term of (6) if we
set
ra
ψp
ra and β as nr and 2α . Even though the proposed
mechanism in Fig.12 can compensate for the coupling of the
2-DOF manipulator, it can be somewhat complicated to
implement. In Subsection VII.C, simplification of the
actuator will be discussed.
ψt
Pretension side
A. Implementation Details
Pretension for
pan wires
Pretension for
tilt wires
Fig.12. Conceptual Design of 2-DOF Manipulator
Therefore we propose a new actuation mechanism that
mechanically compensates for the asymmetric wire
movement, and achieves a proportional relationship between
actuator motion and manipulator motion. Fig.11(b) shows
the basic concept of the proposed actuation mechanism for a
1-DOF manipulator. It is composed of a fan-shaped lever,
and actuator wire pair connected at both ends of the lever.
The length of the actuator wire pair
γ l and γ r
from Fig.11(b) as follows:
π − β +ψ
β ψ
γ l (ψ ) = 2ra sin(
) = 2ra cos( − )
2
2 2
π − β −ψ
β ψ
γ r (ψ ) = 2ra sin(
) = 2ra cos( + )
2
2 2
where
As illustrated in the conceptual designs in Fig.11(b) and
Fig.12, the pivots of the actuation levers must be able to
translate in order to deliver a pretension force to the wires
through the lever mechanism. This moving pivot leads to a
complicated implementation, because the gears and motors
(with associated cabling) connected to the lever must all
move along with the pivot motion. In order to simplify this,
the wire path is redirected as shown in Fig.13(a). Instead of
moving the pivot of the lever, an additional pulley with a
slider at the center moves according to the motion of the
pretension wire (black solid line).
is obtained
(38)
ra , β and ψ denote the radius of the lever, half
of the pan shape angle, and the amount of lever rotation,
respectively. The displacements of the actuator wire pair
9
Fig.14. Detailed Design of 2-module 4-DOF Manipulator
Fig.13. Detailed Design of a 2-DOF Manipulator
Fig.13(b) and (c) show a front and exploded views of the
detailed actuator mechanical design. It is designed to satisfy
the performance specifications in Table II. Two identical
mechanisms are mounted in order to actuate one 2-DOF
manipulator. For actuation of the levers, two 8W BLDC
motors with 53:1 gear heads were used. Between the levers
and the gear heads, 8:1 gear pairs were placed. As the
maximum continuous torque of the motor is 8 mNm and the
efficiency of the gear head is 59%, the approximate
continuous output torque of the lever is 2Nm. In order to
control the wire tensions, 1.22mm pitch lead screws and
springs were used. The tension can be changed from 8.8N to
40N by rotating the lead screws using another geared motor
having the same specification.
As can be seen in Fig.13(b), the wires connected to the
lever are wound three times around pulleys to amplify the
wire motion. In order to minimize friction, miniaturized
bearings were used for the pulleys. The wires connected to
the manipulator shown in Fig.13(d) are also wound three
times to lessen the wire motion and amplify their tension.
This amplification mechanism distributes the stress among
multiple wires, hence, high tension for the manipulator can
be achieved by using low payload, small wires. For
implementation, Polymer wires (Dyneema®) of diameter
0.3mm were used. However, amplifying the tension by using
multiple windings also increases friction, and causes error
between the modeling and implemented system due to
hysteresis in repeated motion. This will be examined in
Section VI.
Fig.14(a) illustrates the assembled structure of a 4-DOF
variable neutral-line manipulator system. Two identical
2-DOF manipulators were stacked, and two identical
actuation mechanisms were connected to them. In order to
transfer wire motion through a flexible path, Teflon conduits
were used. Fig.14(b) shows the assembled structure of the
joint. In order to make the joint exhibit pure rolling, Teflon
flexure rings of 0.5mm thickness were attached between the
links, as shown in Figure 14(b). Most of large frames (made
from ABS material) were fabricated mainly by using 3D
fused deposition manufacturing (FDM). Small parts like
pulleys and sliders were made from aluminum alloy (AL
6061T6).
The diameter of the manipulator was chosen with a large
inner channel to allow passage of multiple dexterous
surgical instruments and an endoscope. Theoretically, based
on the design of the device, there are no strict size
limitations. We expect that 5mm and 3mm diameters are
also feasible, which are standard sizes of conventional
laparoscopic surgical instruments. However, we note that as
the device size shrinks, the payload and the stiffness will
decrease accordingly.
10
Considering practical implementation issues, small-sized
manipulators pose greater fabrication difficulties, such as
fabricating miniaturized flexures and attaching them to
rolling links. We can greatly simplify the mechanism by
removing the flexure and allowing the actuation wires to
replace the function of the flexure, by guiding the rolling
action between the link surfaces. However, the wires cannot
completely prevent slip between the rolling surfaces, which
could lead to inaccuracy in pose control.
The manipulator was designed for multiple uses.
Sterilization is an important issue for such a complicated
mechanism. We are considering low-temperature liquid
sterilization methods because the mechanism has
heat-sensitive components, and thus steam sterilization is not
adequate. For the manipulator shown in Fig.1(b), a cover
sheath made from EPDM has been developed (not shown in
the figure), which is a commonly used material for flexible
endoscopes. In order to prevent the EPDM sheath being
caught between the rolling surfaces and minimize friction
between the sheath and manipulator, a mesh layer made by
woven steel wires was placed between the sheath and the
manipulator. However, the sheath acts to resistant bending
motion and thus slightly increases hysteresis of the
manipulator motion.
VI. EXPERIMENTAL VALIDATION
This section presents experimental results to verify the
proposed manipulator models. The experiments were
conducted using the same specifications as those in Table II.
The stiffness measurement was done on one module by
using a texture analyzer which can measure up to a 49N
maximum force with 0.001N sensitivity.
Fig.15 illustrates the stress-strain curves measured for four
different wire tensions levels. Although the lateral motion is
relatively small compared with the simulation in Fig.10, the
curves are nonlinear and exhibit hysteresis behavior. The
hysteresis is caused by changes in the wire tension according
to the direction of frictional forces acting on the wires. When
the manipulator is driven by a external force, friction aids
the wire tension, and thus the manipulator exhibits higher
stiffness. When the manipulator is driving itself, on the other
hand, friction reduces the wire tension, and thus the
manipulator exhibits lower stiffness.
Sources of friction in the manipulator include a (relatively
small) contribution from the bearings that acompany every
pulley in the actuation mechanism. Moreover, the multiple
windings of the wires in the manipulator not only amplify
the wire tension, but also amplify the friction. Therefore,
using single wire for the manipulator as shown in Fig.6(a),
instead of multiple windings, can reduce the hysteresis.
However, in that scenario, thicker wires and conduits should
be used to a maintain high payload. Applying Teflon coating
or inserting Teflon tubes throughout the wire paths in the
link holes of the manipulator could also reduce the friction
and the resultant hysteresis.
The stiffness change according to the tension variation is
approximately linear as shown in Fig.16 and Table IV. The
simulated stiffness-tension curve is linear, and crosses the
zero point because it has no friction and follows (37). In
contrast, the experimental curve has a nonzero y-intercept of
0.088N/mm. It means that the additional force caused by
friction aids the wire tension.
Fig.17 shows several snapshots of motion of a 4-DOF
variable neutral-line manipulator system prototype. As the
maximum bending angle of one module is more than 45deg,
the manipulator can be bent 90deg as shown in Fig.17(a),(b)
and (c). Under no external force, the motion is not affected
by the stiffness changes, and the motion with low stiffness is
approximately the same as the motion with high stiffness.
)
(N 6
e
c
r
o
F5
)
(N 1.8
e
c
r
o 1.5
F
y = 0.5292x + 0.2276
y = 0.4372x + 0.1915
4
1
y = 0.3339x + 0.0453
3
y = 0.2419x + 0.0839
2
Tension 20N
Tension 33.1N
Tension 46.5N
Tension 59.9N
1
5
10
(a)
15
0.5
0
20
Displacement
(mm)
0
0.5
1
1.5
(b)
2
2.5
3
Displacement
(mm)
Fig.15. Stress-strain graph by experiments
(a) Overall shape of graph (b) Closed-up shape near zero
)
m 0.6
m
/
0.5
N
(
s
s
e 0.4
n
ff
it
S 0.3
0.2
Experiments
Simulations
0.1
0
0
5 10
15
20
Tension (N)
Fig.16. Relationship between wire tension and manipulator
stiffness
TABLE IV
STIFFNESS BY EXPERIMENTS
Tension (N)
20.0
33.1
Stiffness (N/mm)
0.242
0.334
46.5
0.437
59.9
0.529
11
w
(43)
= r sin α = ri sin α i ,
2
where ri and α i are design parameters of an ni -joint
manipulator having the same maximum bending angle and
link width. The stiffness of the
(a)
(b)
ni -joint manipulator is
represented using (42) and (43) as
ri (1 + cosα i )
r sin α (1 + cos( nα / ni ))
(44)
T = Si
T
2
2ni h
2h 2 ni sin( nα / ni )
If we consider a manipulator with a large number of joints
K n i = Si
(c)
(d)
(e)
(f)
r sin α .
(45)
T
nh 2α
For example, the stiffness of the manipulator with an
infinite number of joints is 0.445N/mm when the tension is
59.9N. This value is not severely small compared to the
stiffness of the manipulator with three joints, which is
0.484N/mm. It means that increasing joint number does not
significantly affect the stiffness performance, and we can
select the proper number of joints depending on specific task
requirements.
K ∞ = lim K ni = S∞
ni →∞
Fig.17 2-DOF Manipulator Prototype Performing Controlled
Motions
VII. DISCUSSION
So far, the mechanics of the variable neutral-line
manipulator and the actuation mechanism have been
investigated and verified. In this Section, the effect of joint
number and design variations are discussed.
A. Maximum Bending Angle
As mentioned in Section VI, the maximum bending angle
of one module is approximately 45 deg. This can be
predicted by considering the tension balance. Let us consider
the extreme bending pose of pan motion like Fig.6(b). In this
extreme case, the left pan wire (blue line) has tension T and
the right pan wire (purple line) has zero tension, because the
sum of the tilt wires have tension T and this balances with
the left pan wire. Therefore, the joint will be settled at the
middle of the left pan wire and tilt wire at the center, which
means that the maximum bending angle is half of α . Thus
the maximum bending angle of the module is
θ max =
nα
.
2
(41)
C. Simplification of Actuation Mechanism
In Section V, an actuation mechanism that completely
compensates for the coupling of the 2-DOF manipulator was
proposed. In order to simplify the actuation mechanism, two
1-DOF actuation mechanisms shown in Fig.11(b) can be
used for a 2-DOF manipulator if we can eliminate the
coupling mechanically. Fig.18 introduces a linkage design to
diminish or eliminate the coupling effect. If we enlarge the
space for passage of the center wire, elongation of the center
wire path is diminished or completely eliminated. Fig.18(b)
and (c) shows the linkage with space to the center of the
rolling space, where the length of the wire path in the space
is maintained to
while the original link has additional
length of
as shown in Fig.18(a). However, due to the
removal of
Δ lt
shown in (33), the stiffness decreases as
follows:
F
r cos α
(46)
K = = Sn
T
d
2nh 2
For practical use, this modification will be an efficient
tradeoff between simplicity and performance.
The theoretical maximum bending angle of the
implemented manipulator is 49.8deg, which agrees
reasonably with the measured value.
B. Stiffness Change with Joint Number under Fixed
Maximum Bending Angle and Link Width
Assuming that the maximum bending angle
θ max and the
width of the link w in Fig.4(a) are fixed, we can see that the
stiffness changes as the number of links increases using (37)
and (41). These assumptions can be represented as
θ max =
nα niα i
,
=
2
2
(42)
12
Center of the
rolling surface
Enlarged space
for center wire
d pr
2r
2r
(a)
(b)
(c)
Fig.18.(a) Original Link design, (b) and (c) modified link design
to diminish the coupling between pan and tilt motion
between the wire tension and stiffness of the manipulator,
and its demonstrated stiffness controlling capability without
losing motion control performance. The thin tubular
structure of the manipulator and the ability to control
stiffness by only using position-controlled actuators could
make the manipulator well suited for MIS applications. For
instance, the manipulator can be used as a flexible guide
tube of a single port surgical device. In this application, the
stiffness changing capability can be a great advantage,
because compliance is important for safely moving the guide
tube in the abdominal cavity. On the other hand, high
stiffness is crucial for payload motion or precise operation.
Current work is focusing on integrating all of these efforts
to develop an innovative surgical robotic system, and
evaluating its performance and effectiveness. For future
research, stiffness analysis under arbitrary bending angles
will be investigated. For more precise position control,
friction modeling and analysis deserve study.
REFERENCES
Fig.19. (a) Pre-curved design (b) Rigid-joint like design
D. Design Variation
As mentioned in Section II, the joint shape does not need
to be a cylindrical-shape rolling mechanism. For instance, if
the sum of wire movement has a negative value, we can also
control the stiffness by using pushing shafts instead of using
pulling wire tension. However, in this case the actuation
mechanism should be redesigned to compensate for
asymmetric wire motion.
Fig.19 presents two design variations of the proposed
manipulator. In most cases of snake-like manipulators, the
initial pose is straight shape, which represents a singular
point. As shown in Fig.17(a), the initial pose can be changed
to a C-shape or S-shape by simply changing the individual
link shape. Also, by using different surface angles as
Fig.17(b), a near-rigid joint design is feasible. This
manipulator can be inserted through a curved path by
loosening the wire tension, and can be used like a rigid joint
by applying wire tension.
VIII. CONCLUSION
This paper introduced a unique variable neutral-line
manipulator and actuation mechanism to achieve a
controllable stiffness capability. Design, modeling, analysis
and experimental results of the variable neutral-line
manipulator were presented. Detailed analysis of its stiffness
properties and verification of its performance based on
simulations and experiments proved the direct relationship
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